Re: Z+Size Limitation
zuhair
The concept of size limitation is well known to be applicable in NBG
\MK class theories, but can we have a comparable example in Z related
set theories, here is a trial:
We can add the following size limitation schema to the axioms of Z,
to have the set theory Z+Size Limitation.
Axiom schema of Size Limitation:
If phi(y) is a formula in which at least y is free, and in which x is
not free, then all closures of
~ for all d
( d is ordinal ->
Exist x ( for all y ( y e x -> phi(y) ) and
d equinumerous to x ) )
-> Exist x for all y ( y e x <-> phi(y) )
are axioms.
The idea behind this schema is very clear, if we cannot put
all ordinals that are sets into one-one relation with sets fulfilling
the predicate phi (this is equivalent to saying that we cannot have
an injection from the class of all ordinals that are sets to the
class of all sets fulfilling the predicate phi(in a class theory
admitting proper classes)), then the predicate phi defines a set,
i.e. the class of exactly all sets for which the predicate phi
hold is a set.
According to this schema, all proper classes would be comparable,
i.e. for any two proper classes X,Y there must exist an injection
from X to Y or from Y to X.
However this would also entail that for every class x, if x is not
comparable to the class of all ordinals that are sets, then x is a
set.
Also this schema entails that all sets are either strictly
subnumerous to the class of all ordinals that are sets, or
are incomparable to it.
If this theory is consistent, then it would be maximally
comprehensive, so ZF would no longer be maximally comprehensive.
This theory interpret ZF of course, since clearly replacement is a
theorem schema of it.
I think in this theory we can prove that for any set x, if y is the
class of all sets that are strictly subnumerous to x having every
member of their transitive closures strictly subnumerous to x also,
then y is a set!
Now lets come to a definition that I gave for Cardinality in an
earlier post, let me present it again:
Cardinality(x) is the equivalence class of sets having every member
of their transitive closures strictly subnumerous to x, under
equivalence relation 'bijection'.
Now in this theory we can prove that Cardinality(x) exist for every
set x, and Cardinality(x) of course would be a set!
The nice thing about this theory is that it gives a clear
specification for what a set is and what a proper class is, it
defines a cut off size, compared to which one can know weather a
class is a set or not a set(i.e. a proper class).
So there is a clear demarcation between sets and proper classes in
this theory, and this demarcation is made according to the size
limitation concept, were the size of the class of all ordinals that
are sets is the cut off point of demarcation between sets and proper
classes.
Now if one want to compare between this theory and versions of ZF,
having variable degrees of choice, one can examine the following
alternatives of the size limitation schema, and see their
implication.
(1) If phi(y) is a formula in which at least y is free, and in which
x is not free, then all closures of
Exist s ( for all x ( for all y (y e x -> phi(y)) ->
Exist c (c subclass of s and
x equinumerous to c)))
-> Exist x for all y ( y e x <-> phi(y) )
are axioms.
This version of size limitation is equivalent to weak replacement in
ZF, and the theory Z+ this version of size limitation would be
equivalent to ZF. As I said this theory as ZF doesn't have a clear
demarcation between sets and proper classes.
(2) If phi(y) is a formula in which at least y is free, and in which
x is not free, then all closures of
[for all x ( for all y ( y e x -> phi(y) ) ->
Exist d ( d is ordinal & d equinumerous to x))
and
Exist d ( d is ordinal &
~ Exist x (for all y ( y e x -> phi(y)) and
d equinumerous to x )]
<-> Exist x for all y ( y e x <-> phi(y) )
are axioms.
This will lead to AC ( axiom of choice ) and Replacement.
This version clearly stipulate that all sets are strictly subnumerous
to the class of all ordinals that are sets, thus all sets are well
orderable.
(3) If phi(y) is a formula in which at least y is free, and in which
x is not free, then all closures of
~ for all s Exist x ( for all y ( y e x -> phi(y) ) and
x equinumerous to s )
-> Exist x for all y ( y e x <-> phi(y) )
are axioms.
Now this version of size limitation would lead to global choice and
replacement.
However lets go back to the original theory, does anybody now if it
would be inconsistent.
Zuhair
David Libert
> Hi all,
>
> The concept of size limitation is well known to be applicable in NBG
> \MK class theories, but can we have a comparable example in Z related
> set theories, here is a trial:
>
> We can add the following size limitation schema to the axioms of Z,
> to have the set theory Z+Size Limitation.
>
> Axiom schema of Size Limitation:
>
> If phi(y) is a formula in which at least y is free, and in which x is
> not free, then all closures of
>
> ~ for all d
> ( d is ordinal ->
> Exist x ( for all y ( y e x -> phi(y) ) and
> d equinumerous to x ) )
>
> <-> Exist x for all y ( y e x <-> phi(y) )
>
> are axioms.
I think over ZC (Zermelo's theory with AC) this axiom scheme on phi
is for each phi equivalent to the phi instance of the usual replacement
axiom.
Without AC, just over Z (Zermelo's no AC) it is equivalent (taking
a contrapositve reading) to saying if the class defined by phi
is a propert class then that class embeds each set sized ordinal.
In other words, this last is not claiming so much as the class of
all ordinals injects into the class phi defines. Instead just that
individually each ordinal injects into that class one at a time.
That weaker property I just stated is significant stength, that is
is it not a theorem of Z if Z is consistent. For example, any
Dedekind set is only a set (not a proper class) which doesn't embed
omega.
This property is also strictly weaker than the class of all ordinals
injecting into the phi class (say in NGB models). One way to
describe this, if you make an NGB model up to V_kappa, kappa
an inaccessible, you can have each ordinal individually injecting
into a proper class in the NGB model, if the outer ZF model with
the inaccessible satisified AC out to < kappa choices.
If this outer model has kappa many choice failing, and taking
the NGB model the natural model V_kappa+1 in the obvious way,
you can have a proper class not injecting the class of all ordinals.
> The idea behind this schema is very clear, if we cannot put
> all ordinals that are sets into one-one relation with sets fulfilling
> the predicate phi (this is equivalent to saying that we cannot have an
> injection from the class of all ordinals that are sets to the class of
> all sets fulfilling the predicate phi(in a class theory admitting
> proper classes)), then the predicate phi defines a set, i.e. the class
> of exactly all sets for which the predicate phi hold is a set.
As I was just commenting, I think this property is weaker than injecting
the class of all ordinals unless there is lots of choice (actually
global AC.
> According to this schema, all proper classes would be comparable,
> i.e. for any two proper classes X,Y there must exist an injection from
> X to Y or from Y to X.
I don't think so. Over ZC I think this scheme is equivalent to
replacement, and there are NGB with AC models with diffent
proper class sizes.
> However this would also entail that for every class x, if x is not
> comparable to the class of all ordinals that are sets, then x is a
> set.
I think it says if a class doesn't inject from each ordinal
individually, then it is a set. This is already a non-trivial
property, not a Z theorem (if Z is consistent) but weaker than
injecting the class of all ordinals.
> Also this schema entails that all sets are either strictly
> subnumerous to the class of all ordinals that are sets, or
> are incomparable to it.
If I am correct above that this schematum is equivalent over
ZC to replacemernt, with AX this theoetry is equivalrnt to ZFC,
and if ZFC is consistent it has models in qhich every proper
class is comparable to the calls of all ordinals, in fact injects
it. For example, global choice models wherre all proper classes
are equipollent. Or as Aatu noted, models with many proper
class sizes but prper classes injecting the class of all ordinals
by GCH failures at the inaccessible.
> If this theory is consistent, then it would be maximally
> comprehensive, so ZF would no longer be maximally comprehensive.
But over AC, if I am right it is just equivalent to ZFC.
> This theory interpret ZF of course, since clearly replacement is a
> theorem schema of it.
With AC this axiom get replacement as you said. Without AC I think
not. You could have Z models with non-wellorderable sets and replacement
failing over these, but all casses of replacement holding over
well-orderable sets. Then the axiom above would hold while replacement
didn't.
I had been thinking of the early sections above, and had enough
thoughts about them to start with post.
The sections below, I hadn't thought about as much, and I guess
its better for me not to try to wing it woth these.
So I won't comment further on the parts below for this post.
> This would be equivalent to AC ( axiom of choice ).
>
> This version clearly stipulate that all sets are strictly subnumerous
> to the class of all ordinals that are sets, thus all sets are well
> orderable.
>
>
> (3) If phi(y) is a formula in which at least y is free, and in which x
> is
> not free, then all closures of
>
> ~ for all s Exist x ( for all y ( y e x -> phi(y) ) and
> x equinumerous to s )
>
> <-> Exist x for all y ( y e x <-> phi(y) )
>
> are axioms.
>
> Now this version of size limitation would lead to global choice.
>
> However lets go back to the original theory, does anybody now if it
> would be inconsistent.
>
> Zuhair
--
David Libert ah...@FreeNet.Carleton.CA
zuhair
> zuhair (zaljo...@gmail.com) writes:
> > Hi all,
>
> > The concept of size limitation is well known to be applicable in NBG
> > \MK class theories, but can we have a comparable example in Z related
> > set theories, here is a trial:
>
> > We can add the following size limitation schema to the axioms of Z,
> > to have the set theory Z+Size Limitation.
>
> > Axiom schema of Size Limitation:
>
> > If phi(y) is a formula in which at least y is free, and in which x is
> > not free, then all closures of
>
> > ~ for all d
> > ( d is ordinal ->
> > Exist x ( for all y ( y e x -> phi(y) ) and
> > d equinumerous to x ) )
>
> > <-> Exist x for all y ( y e x <-> phi(y) )
>
> > are axioms.
>
> I think over ZC (Zermelo's theory with AC) this axiom scheme on phi
> is for each phi equivalent to the phi instance of the usual replacement
> axiom.
>
> Without AC, just over Z (Zermelo's no AC) it is equivalent (taking
> a contrapositve reading) to saying if the class defined by phi
> is a propert class then that class embeds each set sized ordinal.
This needs clarification, what do you exactly meant by "embeds".
>
> In other words, this last is not claiming so much as the class of
> all ordinals injects into the class phi defines. Instead just that
> individually each ordinal injects into that class one at a time.
But ordinals are nested transitive sets! I think the result would be
claiming so much as the class of all ordinals injects into the class
phi defines.
We seem to differ on this point, so please can you clarify in details
with examples, because this is the essential point.
Zuhair
zuhair
To further clarify matters, let us work in MK, for a moment.
Lets take the axiom of size limitation in MK to be the following:
For all x ( ~ Exist f (f:D->x, f is injective) -> x e V )
were V={y| y is a set}
so V is the class of all sets, which is a proper class of course.
were D={y| y is ordinal & y is a set}
i.e. D is the class of all ordinals that are sets, which is a proper
class of course.
So what this axiom is saying is that for every set x is incomparable
to D, or if x is strictly subnumerous to D, then x is a set.
Of course the oppose direction i.e.
For all x ( x e V -> ~ Exist f (f:D->x, f is injective) )
is already provable in MK.
so we have
For all x ( x e V <-> ~ Exist f (f:D->x, f is injective) )
So every class x is a set if and only if either x is strictly
subnumerous to D, or
x is incomparable to D.
In this way all proper classes would be comparable to D.
Now can the above be true, i.e. is there a problem with the above.
If there is no problem with the above, i.e. there is no inconsistency
involved with the above size limitation axiom, then I wanted to fined
a comparable way of doing the above but with a "set" theory, i.e. a
theory that do not permit proper classes to be objects in its domain
of discourse, that what made be write the above axiom.
In a *set* theory we can find a comparable way by using *formulas*
So we say for example that the formula phi defines a proper class if
the following is fulfilled:
for all d (d is ordinal -> Exist x (for all y ( y e x -> phi) & x
equinumerous to d))
and we say that if the negation of the above is true, then phi would
define a set.
So to iterate the size limitation schema again:
Axiom Schema of Size Limitation:
If phi(y) is a formula in which at least y is free, and in which x is
not free, then all closures of
~ for every ordinal d Exist x (for all y ( y e x -> phi) & x
equinumerous to d)
-> Exist x for all y ( y e x <-> phi(y) )
are axioms.
Of course the above schema is to be added to Z, and the term
'equinumerous'
is defined as in Z (Zermelo set theory).
I think that this is equivalent to the following axiom in MK(Morse-
Kelley):
For all x ( ~ Exist f (f:D->x, f is injective) -> x e V ).
So we are confonted with two questions:
(1) is the general approach of size limitation in this manner
consistent?
(2) are these two approaches equivalent?
Zuhair
Aatu Koskensilta
> I think that this is equivalent to the following axiom in MK(Morse-
> Kelley):
>
> For all x ( ~ Exist f (f:D->x, f is injective) -> x e V ).
Why? Provably in MK there are proper classes that are not
(parametrically) definable in the first-order language of set theory.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
zuhair
> zuhair <zaljo...@gmail.com> writes:
> > I think that this is equivalent to the following axiom in MK(Morse-
> > Kelley):
>
> > For all x ( ~ Exist f (f:D->x, f is injective) -> x e V ).
>
> Why? Provably in MK there are proper classes that are not
> (parametrically) definable in the first-order language of set theory.
I need some clarification regarding this issue, please, what did you
exactly wanted to say. did you mean definable as objects in a *set*
theory, then all proper classes are not definable in *set* theories.
(I use the term *set* theory to mean a theory that do not allow proper
classes to be objects in their universe of discourse, as opposed to
*class* theories which allows that). Or you meant something else.
Possibly you mean proper classes that are not definable by any formula
phi of first order logic language.
For example we take the formula "x is ordinal" to be a proper class
defining formula in Z set theory, and this would be comparable to D in
Morse-Kelley.
But perhaps you mean that there is a proper class X in Morse-Kelley,
such that there is no formula phi in FOL such that X={y|phi}, perhaps
that what you meant.
hmmm..., if that what you meant, then this will not affect sets, it
would affect proper classes, however all sets in MK(with size
limitation as above) would be so in Z+size limitation, but not all
proper classes in MK(with size limitation as above) can be defined in Z
+size limitation. Possibly I am mistaken since MK allow quantification
over proper classes, possibly what I just said is right of NBG with
the above size limitation axiom but not of MK.
Zuhair
Zuhair
>
> --
> Aatu Koskensilta (aatu.koskensi...@uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
Aatu Koskensilta
> But perhaps you mean that there is a proper class X in Morse-Kelley,
> such that there is no formula phi in FOL such that X={y|phi}, perhaps
> that what you meant.
Yep, there are, provably in MK, proper classes X such that X is not {x |
P(x, a1,..., an)} for any formula P in the first-order language of set
theory and sets a1, ... , an. An example of such a class is the class of
all true sentence in the first-order language of set theory with
parameters.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
zuhair
> zuhair <zaljo...@gmail.com> writes:
> > But perhaps you mean that there is a proper class X in Morse-Kelley,
> > such that there is no formula phi in FOL such that X={y|phi}, perhaps
> > that what you meant.
>
> Yep, there are, provably in MK, proper classes X such that X is not {x |
> P(x, a1,..., an)} for any formula P in the first-order language of set
> theory and sets a1, ... , an. An example of such a class is the class of
> all true sentence in the first-order language of set theory with
> parameters.
>
Yea, you are right, but that would not impose a great problem with my
approach generally. Really, I know that this would always be an issue,
but what I can say here is that this size approach that I did in *set*
theories is the nearest to the one in MK and comparable class
theories ,though it will not be equivalent to it, I think this is the
most we can get with size limitation modified to *set* theories.
Regards.
Zuhair
> --
> Aatu Koskensilta (aatu.koskensi...@uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
zuhair
'Z+Size Limitation' is the set of all sentences entailed (from First
Order Logic with Identity ' =', and Epsilon membership 'e' ) by the
following non logical axioms:
1) Extensionality: for all z ( z e x <-> z e y ) -> x=y
2) Regularity:
For all x ( Exist y (y e x) ->
Exist z ( z e x & ~ Exist c (c e z & c e x)))
3) Separation: if phi(y) is a formula in which at least y is free,
and in which x is not free then all closures of
For all c Exist x for all y ( y e x <-> ( y e c & phi(y) ) )
are axioms.
4) Pairing: For all z,y Exist x ( z e x & y e x )
5) Union: For all c Exist x for all z,y ((z e y & y e c) -> z e x)
6) Power: For all c Exist x for all y
(for all z (z e y -> z e c) -> y e x)
7) Infinity: Exist N (0 e N & for all x (x e N -> xUnion{x} e N)).
8) Size Limitation:If phi(y) is a formula in which at least y is
free, and in which x is not free, then all closures of
~ for every ordinal d Exist x ( for all y (y e x -> phi(y)) &
d equinumerous to x )
<-> Exist x for all y ( y e x <-> phi(y) )
are axioms.
were d equinumerous to x is defined in the standard manner.
Theory definition finished/
Now the question that present itself, is this theory equivalent to
ZF?
We know for sure that weak replacement is provable in this theory,
and since this theory already contain pairing and separation
then we only need to prove weak replacement in order to have
ZF as a sub-theory of this theory, since the later is proved, then
ZF is a sub-theory of this theory.
Is the axiom schema(8) provable in ZF? If so
then this theory would be a sub-theory of ZF,
thereby both theories would be equivalent.
So we have : Z+SL -> ZF
But I really doubt that we have ZF -> Z+SL
were "SL" stands for "Size Limitation".
Regards
Zuhair
David Libert
Here are the various things I think are true, related to all this.
AC is the usual axiom of choice: set many set sized choices.
Let GC Global choice: be class many set sized choices
(phrased in NGB).
NGB is usually axiomatized to include GC. But if you replace GC
by AC in NGB, then if the resulting theory is consistent it
doesn't prove GC.
Basically: GC is strictly stronger than AC.
Consider proper classes in a model of variations of NGB but
with all AC possibly removed. (ie we consider no AC, AC only
or GC as various cases of interest).
We put a partial pre-order on proper clasess by the existence of
an injection (which woild be a proper class of 2 typles).
What are the possible partial orderings on these mod equipollence?
There is exactly one class up to equipollence, ie all proper
classes are equipollent ("same size") <-> GB.
So let's consider what can happen in ~GB models, ie at least 2
sizes of proper classes.
V is the class of all sets. It is largest in the ordering,
everything injects into it.
Let D be the class of all set ordinals.
D is minimal in the partial orderring. No proper class is
strictly below D.
It is possible to have models where D is in fast smallest:
every proper class is above D.
It is possible to have other models where there are incomparables
to D.
These are even possible as AC models. Not GC models of course,
because then all proper classes are equipollent.
In either case, D smallest or else incomparables to D, it is
posssible to have many different sizes. In fact as many as V
many different sizes.
All these are possible in AC + ~GC models.
If you go to ~AC models, you can get an even stronger form of
incomparabilty, beyond incomparable to D as above.
To state these we must go beyond ther language of just order
relations on proper classes.
In ~AC models, it is possible to have proper classes that are
incomarable to individual set sized ordinals. The most extreme
cases are proper classes incomparable to omega, which can
happen in models without countable choice.
Now to relate these to the properties above.
Recall:
> 8) Size Limitation:If phi(y) is a formula in which at least y is
> free, and in which x is not free, then all closures of
>
> ~ for every ordinal d Exist x ( for all y (y e x -> phi(y)) &
> d equinumerous to x )
>
> <-> Exist x for all y ( y e x <-> phi(y) )
I will call this SL, as Zuhair did above.
SL is equivalent to the statement that every proper class is
> each set sized ordinal. So in other words, we are not in the
last case as above.
A stronger statement is D is smallest proper class: each
proper class is > D.
D is smallest -> SL.
But over NGB with GB replaced by AC, there is no proof of
SL -> D is smallest, if that theory is consistent.
> So we have : Z+SL -> ZF
I agree.
> But I really doubt that we have ZF -> Z+SL
I agree. I think if ZF is consistent it doesn't prove SL.
On the other hand : ZFC proves SL.
So without AC : SL -> replacement and only that direction.
But over Z + AC : SL and replacement are equivalent.
So what about Zuhair's definitions of cardinality? I will consider
the recent versions.
One of these was Cardinality(x) = the class of all sets equinumerous
to x with all members of their transitive closures strictly subnumerous
to x.
A second recent version was the class of all well-founded sets equipollent
to x with all members of the transitive closure strictly subnumerous to x
of smallest rank for such sets.
One question is for x is set is Cardinality(x) a set.
I think there is a ZF proof of this, Cardinality(x) is a set
using regularity and replacement but not needing any AC.
But regularity and replacement are essential for the first definition.
There is a conterexample model dropping regularity and keeping replacement,
and another counterexample model keeping regularity and dropping replacement.
These both assuming ZF is consistent.
Or else we can also replace regularity by the acioms Zuhair mentioned
recently: every set is equipollent to a well-founded set. Using this
axiom and replacement, the second definition above produces a set,
no AC needed.
But there is another prpblem for both definitions. I posted about
this in one of the recent threads.
If ZF is consistent there are ZF models with non-isomorphic amorphous
sets. These have the same cardinality, using either definition uniformly.
They shouldn't have the same cardinality though, for a reasonable
definition of cardinality, since they are not isomorphic.
And I think from the consistency of ZF we can get a model of
ZF + there are 2 non-isomorphic amorphous sets + SL.
So this even makes a problem for the 2nd definition with
its background assyumption.
On the other hand, over ZFC all is ok. With AC both definitions
have the property that non-isomorphic sets really do get assigned
different cardinalities.
So the definitions work well over ZFC.
--
David Libert ah...@FreeNet.Carleton.CA
zuhair
Yea, I thought of that but this would be equivalent to the above (see
below).
There is another definition I don't know if I mentioned it though:
Cardinality(x) is the class of all hereditarily hereditary sets
equinumerous to x.
were a hereditary set is a set were every member of its transitive
closure is strictly subnumerous to it.
and a hereditarily hereditary set is a set of hereditary sets strictly
subnumerous to it.
> One question is for x is set is Cardinality(x) a set.
>
> I think there is a ZF proof of this, Cardinality(x) is a set
> using regularity and replacement but not needing any AC.
You think so, what is that proof, you never mentioned it fully.
>
> But regularity and replacement are essential for the first definition.
> There is a conterexample model dropping regularity and keeping replacement,
> and another counterexample model keeping regularity and dropping replacement.
> These both assuming ZF is consistent.
>
> Or else we can also replace regularity by the acioms Zuhair mentioned
> recently: every set is equipollent to a well-founded set. Using this
> axiom and replacement, the second definition above produces a set,
> no AC needed.
>
> But there is another prpblem for both definitions. I posted about
> this in one of the recent threads.
>
> If ZF is consistent there are ZF models with non-isomorphic amorphous
> sets. These have the same cardinality, using either definition uniformly.
Do you mean even with Regularity?
The way I used to see matters is that if Regularity is assumed, then
this definition work, and there is no possibility for having two sets
not equinumerous having the same cardinality or the opposite.
If Regularity is assumed then one can prove that
for all x there exist y
(y equinumerous to x & for all m ( m e TC(y) -> m strictly subnumerous
to x)
That is easy, since for any set A there is a set B that is
equinumerous to A with the minimal rank, i.e. sets in all ranks of the
cummulative hierarchy below that rank in which B exist, would be sets
that are strictly subnumerous to A. Now since B has the minimal rank,
and since B is a subset of its rank in the cummulative hierarchy, then
all members in B would be strictly subnumerous to B and thus A, right!
which proves the above theorem.
So actually I think that what is called "Scott cardinal" is in reality
a subset
of the cardinals I defined , so we have
for all x ( Scott cardinality (x) subset of Z.cardinality(x) )
Since assuming Regularity we have
for all x ( Scot cardinality (x) is non empty )
Then Z.Cardinals are always non empty for all sets!
However I thought that the on-isomorphic amorphous sets that you were
speaking of, come to surface if we do not assume Regularity, that's
what I though, am I right or wrong regarding this point?
Zuhair
David Libert
> On Dec 5, 2:03=A0am, ah...@FreeNet.Carleton.CA (David Libert) wrote:
[Deletion]
>> =A0 So what about Zuhair's definitions of cardinality? =A0 I will conside=
> r
>> the recent versions.
>>
>> =A0 One of these was =A0 Cardinality(x) =A0=3D =A0the class of all sets e=
> quinumerous
>> to =A0 x =A0with all members of their transitive closures =A0strictly sub=
> numerous
>> to x.
>>
>> =A0 A second recent version =A0was the class of all well-founded sets equ=
> ipollent
>> to x =A0with all members of the transitive closure strictly subnumerous t=
> o x
>> of smallest rank for such sets.
>
> Yea, I thought of that but this would be equivalent to the above (see
> below).
That second definition was to work when we drop regularity and replace
it with the axiom that every set is equipllent to some well-founded set.
What do you mean by equivalence of the two definitions?
In general, the first one will be a bigger class. There can be higher
rank sets than minimal in the first definition. And in the models
we mentioned dropping regularity the first definition can be a proper
class. The 2nd definition will always be a set for models of the
axiom every set is equipollent to some well-founded set.
> There is another definition I don't know if I mentioned it though:
>
> Cardinality(x) is the class of all hereditarily hereditary sets
> equinumerous to x.
>
> were a hereditary set is a set were every member of its transitive
> closure is strictly subnumerous to it.
>
> and a hereditarily hereditary set is a set of hereditary sets strictly
> subnumerous to it.
I think this definition was not stated before, though another similar
one was.
To collect here all definitions in one place:
Your early threads gave 2 first definitions. In
[1] David Libert "The magic of Hereditarily Hereditary Cardinals"
sci.logic, sci.math Nov 29, 2009
http://groups.google.com/group/sci.math/msg/1b40b261aeff6e96
I gave exact references to your articles giving these 2 definitions
and quoted them.
I quote those 1st 2 definitions from [1] :
First definition:
>I would like to suggest the following definition:
>4) The cardinality of any set x is: The class of all sets
>that are equinumerous to x were every member of their transitive
>closure is strictly subnumerous to x.
Second definition:
> "A cardinal is a class of all hereditarily hereditary sets strictly
> subnumerous to some set".
I quoted you above quoting me repeating your 3rd and 4th definitions
Then I quoted you giving a new 5th definition.
In [1] I introduced a uniform notation for definitions like this.
Your 2nd definition had signatue < _ < in that notation. The new
5th definition has signature = _ < .
>> =A0 One question is =A0for x is set =A0is Cardinality(x) =A0a set.
>>
>> =A0 I think there is a ZF proof of this, =A0Cardinality(x) =A0is a set
>> using =A0regularity and replacement =A0but not needing any AC.
>
> You think so, what is that proof, you never mentioned it fully.
You are right, I claimed Jech's argument should generalize but never wrote
out details.
I tried to prepare the details to answer you here and I found a flaw.
I think my previously claimed generalization of Jech's ZF proof that
HC is a set does generalize as I previously claimed to H_kappa
for kappa von Neumann cardinals.
But I overlooked that I was using x is well-orderable in the proof.
So I retract that claim for now.
So we are back to as you say: we don't know if these cardinals are
sets over just ZF.
>> =A0 Or else =A0we can also replace regularity by the acioms Zuhair mentio=
> ned
>> recently: =A0every set is equipollent to a well-founded set. =A0Using thi=
> s
>> axiom and replacement, =A0the second definition above produces a set,
>> no AC needed.
This would be the 4th definition above, a combionation of Scott's trick
and Zuhair's definition. This claim I made still stands, since the
Scott's trick part of the definition does it. This doesn't depend on
generalizing Jech's proof.
>> =A0 But there is another prpblem for both definitions. =A0I posted about
>> this in one of the recent threads.
>>
>> =A0 If ZF is consistent there are ZF models with =A0non-isomorphic amorph=
> ous
>> sets. =A0These have the same cardinality, using either definition uniform=
> ly.
>
>
> Do you mean even with Regularity?
Yes. I have the proof of this for definitions #1 and #2 in [1].
The other definitions would be similar.
> The way I used to see matters is that if Regularity is assumed, then
> this definition work, and there is no possibility for having two sets
> not equinumerous having the same cardinality or the opposite.
>
> If Regularity is assumed then one can prove that
>
> for all x there exist y
> (y equinumerous to x & for all m ( m e TC(y) -> m strictly subnumerous
> to x)
>
> That is easy, since for any set A there is a set B that is
> equinumerous to A with the minimal rank, i.e. sets in all ranks of the
> cummulative hierarchy below that rank in which B exist, would be sets
> that are strictly subnumerous to A. Now since B has the minimal rank,
> and since B is a subset of its rank in the cummulative hierarchy, then
> all members in B would be strictly subnumerous to B and thus A, right!
> which proves the above theorem.
You are right aboit the minimal rank. That is ZF, no AC needed. It
uses regularity as you are assuming, so that is all ok.
Since B is of minimal rank to be equinumerous to A, all members of
B and indeed of Tc(B) would not be equinumerous to A.
But in ZF without choice, there are 2 ways a set can fail to be
equinumerous or supernumerous to A. It can be strictly subnumerous
to A, as you note, or it can be incomparable to A.
That incomparability case can't happen in ZFC.
In fact over ZF, all sets being comparable is actually equivalent
to AC.
So your next step, that all members of B must be strictly subnumrous
to B is unwarrented. Some might be incomparable to B.
Aatu recently noted a similar point.
My [1] proof was also related to this.
> So actually I think that what is called "Scott cardinal" is in reality
> a subset
> of the cardinals I defined , so we have
>
> for all x ( Scott cardinality (x) subset of Z.cardinality(x) )
This would be true over ZFC, where not-supernumerous -> strictly
subnumerous and your argument above is ok.
If there are more than Beth_2 many non-isomorphic sets
(as can be arranged in a Cohen model), then they can't all have
Scott cardinaal of rank <= omega+1. (There are only Beth_2 many sets
of rank <- omega+1).
So for A amorphous with Scott cardinal > omega, find a member
b of the Scott cardinal of A with rank(b) >= omega+1. Replace some
member of b ny omega, producing b'. The b' has same cardinality
and rank as b, so b' is in the Scott cardinal of A also.
So omega is in TC(Scott cardinal(A)). But omega is not subnunerous
to A since A is amorphous.
So Scott cardinal(A) differs from your definitions based on
hereditarty.
Opps, for heriditarily hereditary take an extra Beth level and
in b put in an omega 2 membership levels down not one.
> Since assuming Regularity we have
>
> for all x ( Scot cardinality (x) is non empty )
>
> Then Z.Cardinals are always non empty for all sets!
As noted above they can diverge. See also the proof in [1].
> However I thought that the on-isomorphic amorphous sets that you were
> speaking of, come to surface if we do not assume Regularity, that's
> what I though, am I right or wrong regarding this point?
>
> Zuhair
The Cohen models get amorphous sets in ZF models including
regularity.
--
David Libert ah...@FreeNet.Carleton.CA
zuhair
> zuhair (zaljo...@gmail.com) writes:
> > On Dec 5, 2:03=A0am, ah...@FreeNet.Carleton.CA (David Libert) wrote:
>
I need to look into this carefully, and actually I need to review all
your replies carefully really.
But, for the time being, I have a question: these amorphous sets that
would have the same cardinals (I defined) though not equinumerous, is
it the same case with scott cardinals or not? I mean my definition
failed with these sets, does scott cardinals also fail with these sets
or not?
Zuhair
David Libert
> On Dec 6, 3:13=A0am, ah...@FreeNet.Carleton.CA (David Libert) wrote:
>> zuhair (zaljo...@gmail.com) writes:
>>
>
> I need to look into this carefully, and actually I need to review all
> your replies carefully really.
>
> But, for the time being, I have a question: these amorphous sets that
> would have the same cardinals (I defined) though not equinumerous, is
> it the same case with scott cardinals or not? I mean my definition
> failed with these sets, does scott cardinals also fail with these sets
> or not?
>
> Zuhair
The Scott definition is ok for these cases. It is a theorem of ZF
that the Scott definition assigns different Scott cardinalities to
non-isomorphic sets, so it works for all cases.
With regularity, the Scott-cardinal(x) always contains members
isomorphic to x.
So if x1 is not isomorphic to x2, then Scot-cardinal(x1)
will contain members isomorphic to x1.
Since x1 is not isomorphic to x2, these Scott-cardfinal(x1)
members won't be isomorphic to x2.
But every memeber of Scott-cardinality(x2) is isomorphic
to x2, so those Scott-cardinality(x1) members won't be
members of Scott-cardinailty(x2).
So for x1, x2 not isomorphic, Scott-cardinality(x1)
and Scott-cardinality(x2) are distinct.
--
David Libert ah...@FreeNet.Carleton.CA
zuhair
>
> - Show quoted text -
Thanks a lot David.
I have presented another definition of Cardinality in a separate
topic, I don't know if it will overcome these Amorphous sets.
However there is one thing that puzzles me really and that is, as you
know I defined this theory, I mean Z+SL in order to prove that the
class of all sets hereditarily strictly subnumerous to a set is a
set, however the proof still eludes me, it is still not clear to me
even in this rather strong theory that the class of
all sets hereditarily strictly subnumerous to some set
is a set?
Seeing all this unclarity I think I need to either simply axiomatize
that the class of all sets hereditarily strictly subnumerous to a set
is a set, or follow that last resort which may be erroneous, which is
the following:
The basic idea is
---No proper class has all members of its transitive closure strictly
subnumerous to some set. ----
Of course I am assuming both Regularity and Strong Extensionality:
Strong Extensionality is a strong version of axiom of Extensionality
we define "proper member" in the following manner:
y is a proper member of x <-> ( y e x & ~ y = x )
Strong Extensionality simply states:
All sets with the same proper members are identical.
In symbols:
For all z ((z e x & ~ z=x) iff (z e y & ~ z=y)) -> x=y
So assuming strong Extensionality and Regularity, I think that
all proper classes cannot have all members of their transitive
closures strictly subnumerous to some set.
In symbols:
For every proper class x ~ Exist s for all y
( y e TC(x) -> y strictly subnumerous to s))
Actually the above could be also written in a *set* theory, but the
formulation would be rather complex, it would be something like the
following schema:
if phi(y) is a formula in which at least y is free, and in which x is
not free, then all closures of
Exist s for all x,z ((for all y ((y e x -> phi(y)) & z e TC(x)) ->
z strictly subnumerous to s)
-> Exist x for all y ( y e x iff phi(y) )
are axioms.
Of course this axiom schema if proves to be not inconsistent with the
other axioms of Z+SL, or even Z alone, then it easily proves that the
class of all sets hereditarily strictly subnumerous to x is a set( it
means the class of all sets that are strictly subnumerous to a set x
having every member of their transitive closures also strictly
subnumerous to x), since all sets in the transitive closure of that
class would be subnumerous to x, thus from the schema above
this class is a set!
I would welcome any help, since the above is very doubtful.
Does anybody know if a proper class that has all members of its
transitive closure strictly subnumerous to some set?
Zuhair